Previous page: Mathematical models of scattering 

George Biddell Airy's paper ^{(1)} entitled "On the intensity of light in the neighbourhood of a caustic"was published in 1838. In this paper, he showed that the intensity of light in a rainbow could be modelled using a cubic wavefront. He provided a table of values of what is now universally known as the "Airy Integral" and wrote:
"The extent of this table for the positive values of m is not so great as I would wish; but it goes far enough to enable us to point out the most remarkable circumstances of the distribution of illumination."The rather wistful first part of this remark suggested that Airy had devoted substantial personal effort to evaluating his integral. With the aid of modern computers, calculations using Airy theory have now become almost trivial. Despite the development of much more complex mathematical models, Airy theory remains a useful tool. Although it was originally developed as a model for the primary rainbow (p = 2), Airy theory has been extended to apply to arbitrary values of p ^{(2,3)} and to deal with both perpendicular and parallel polarisations.^{(4)}
Fig. 1 Comparison of p = 2 rainbow calculated by
Mie theory and Airy theory r = 103.4057 µm, λ = 0.65 µm, x = 2 π r / λ = 1000 n = 1.333, perpendicular polarisation 
Fig. 1 compares Mie theory and Airy theory for the primary rainbow for x = 1000 and n 1.333.
Fig. 1 demonstrates that Airy theory provides a good approximation
of the general features of the primary rainbow, such as the broad maximum
around 139° and the maxima of the supernumerary arcs around 141.1°,
142.6° and so on. The ripples on the Mie theory calculations
are due to interference between the p = 2 rainbow rays (which have
suffered one internal reflections within the sphere) and p = 0 rays (which are
reflected from the exterior of the sphere)  as explained here.
As the calculations using Airy theory only concern the effects of p = 2
rays, it is perhaps rather unfair to complain that Airy theory does not
reproduce the ripples predicted by Mie theory. It is probably
fairer to compare Airy theory with calculations using Debye series for a
specific value of p (such as p = 2 for the primary rainbow)  as shown below in
Fig. 2.
Fig. 2 Comparison of p = 2 rainbow
calculated by
Airy theory and Debye series r = 103.4057 µm, λ = 0.65 µm, x = 2 π r / λ = 1000 n = 1.333, perpendicular polarisation 
Fig. 2 shows that Airy theory agrees very closely with the results provided by the rigorous Debye series  despite the relative simplicity of the Airy calculations. However, Airy theory miscalculates the scattering angles for the maxima and minima of the supernumerary arcs above 142°.
Note that Figs. 2  6 on this page are equivalent to Figs. 3  7 in the paper by Hovenac and Lock. ^{(3)}
Fig. 3 Comparison of p = 3 rainbow
calculated by
Airy theory and Debye series r = 103.4057 µm, λ = 0.65 µm, x = 2 π r / λ = 1000 n = 1.333, perpendicular polarisation 
Fig. 3 extends the comparison to the secondary rainbow (p = 3). In this case, Airy theory is quite accurate in predicting the principal maximum around 127.5°, but much less accurate for the supernumerary
arcs  in terms of scattering angles and intensity.
Fig. 4 Comparison of p = 4 rainbow
calculated by
Airy theory and Debye series r = 103.4057 µm, λ = 0.65 µm, x = 2 π r / λ = 1000 n = 1.333, perpendicular polarisation 
Similar comparisons of Airy theory and the Debye series for higher
order rainbows (p = 4, 5 and 6) are shown in Figs. 4  6 respectively.
In each case, Airy theory seems adequate for calculation of the
principal maximum, but much less accurate for the supernumeraries.
Fig. 5 Comparison of p = 5 rainbow
calculated by
Airy theory and Debye series r = 103.4057 µm, λ = 0.65 µm, x = 2 π r / λ = 1000, n = 1.333 perpendicular polarisation 
Fig. 6 Comparison of p = 6 rainbow
calculated by Airy theory and Debye series r = 103.4057 µm, λ = 0.65 µm, x = 2 π r / λ = 1000, n = 1.333 perpendicular polarisation 
Of course, "monochromatic" rainbows caused by illumination of a
sphere by light of a single wavelength do not correspond to the popular notion
of coloured rainbows. It is therefore appropriate to examine the
performance of Airy theory in predicting natural rainbows (i.e. those
due to the scattering of sunlight). Figs. 7 and 8 show the results of
calculations using Mie theory and Airy theory.
r = 100 µm, sunlight 
r = 100 µm, sunlight 
Comparison of Figs. 7 and 8 confirms that Airy theory can be successfully used to simulate rainbows caused by sunlight, but also reveals several key differences:
r = 100 µm, sunlight 
Fig. 10 Lee diagram calculated using Mie theory  Fig. 11 Lee diagram calculated using Airy theory 
Comparison of Figs. 10 and 11 shows very close agreement between the results obtained with Mie theory and Airy theory. Nevertheless, there are some subtle differences, such as the dark gaps between the supernumerary arcs (e.g. near 142° for r = 50 µm) being darker and more clearly defined when using Airy theory. Fig. 11 is not as accurate than Fig. 10  but Fig. 11 was computed in less than 2 hours, whilst Fig. 10 took almost 1 week!
Airy theory can be successfully used to model scattering in the vicinity of rainbow angles  at least for the primary (p = 2) and secondary (p = 3) rainbows. However, Mie theory is essential for more complicated scattering mechanisms, such as those causing the glory.
MiePlot offers the option of calculations based on:
Page updated on 23 April 2010
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